104 XIII INTEGRATION A measure /<( on X is said to be real if p = //, or equivalently if jLi(f) is real for every/e Jf R(X). We may therefore identify the set of real measures on X with a vector space of linear forms on the real vector space JTR(X). This vector space is denoted by MR(X). Lebesgue measure and all Dirac measures are real. If /JL is any complex measure, then the measures /^ = (/* + /I)/2 and ju2 = GU — fi)/2i are real. They are called respectively the real and imaginary parts of ju, and are denoted by 0t\i and J\i respectively. For each function /e JTR(X), we have (13.2.1) and by definition (13.2.2) 3. POSITIVE MEASURES: THE ABSOLUTE VALUE OF A MEASURE A measure n on a locally compact space X is said to be positive if, for each function fe ^TR(X) such that /^ 0, we have //(/) ^ 0. Consequently, if /and g are two functions belonging to JTR(X) such that/^^, we have X/) ^ /((#). Since each/e «#*R(X) can be written in the form /=/+ -/" (where f+(x) = (/(x))+ and /"» = (/(x))™ (2.2)), it follows that a posi- tive measure is a raz/ measure. We denote by M + (X) the set of all positive measures on X. Surprisingly, the property of positivity alone implies the defining property (13.1.2) of a measure: (13.3.1) Let n be a linear form on the real vector space «3fR(X) such that //(/) *> 0 whenever /^ 0. Then /i is a (positive) measure on X. We have to show that (1 3.1 .2) is satisfied. There exists a function g e «^R(X) with values in [0, 1] and equal to 1 throughout K ((3.18.2) and (4.5.2)). Hence for all/e <?TR(X; K), we have and therefore o ^ jjL(f+) z \\f\\ • fa), o g x/-) ^ ll/ll so that finally |X/)I^ 2 ||/|| by /L We have Ji = ^, and if A, ^ are measures on X and a, b are